First-order buoyancy correction of modal instabilities in stratified boundary layers

TL;DR

This work develops a perturbation framework to quantify buoyancy effects on modal instabilities in stratified boundary layers within a fully compressible, non-Oberbeck-Boussinesq formulation. By treating the Richardson number $Ri$ as a small parameter and using an adjoint–residual approach, it derives a first-order eigenvalue correction $\alpha = \alpha_0 + C_0 Ri$ that can be computed from the neutrally buoyant problem alone. The method accurately predicts shifts in neutral curves, growth rates, phase speeds, and $N$-factors for ideal gases across a wide range of $Pr$, $M$, and wall–free-stream temperature ratios, with a strong dependence on $Pr$. For non-ideal, supercritical fluids, local evaluation of $C_0$ near sharp property gradients (pseudo-boiling) remains essential, yet the OB contribution still dominantly controls buoyancy effects, enabling reliable transition predictions through the first-order correction. Overall, the framework offers an efficient, scalable tool to assess buoyancy-driven modifications to boundary-layer instabilities and transition in both standard and extreme thermophysical regimes.

Abstract

We present a perturbation-based framework that captures buoyancy effects on modal instabilities in stratified boundary-layer flows within the fully compressible, non-Oberbeck-Boussinesq formulation. Treating the Richardson number as a small parameter and recasting the stability problem into an adjoint-residual form, we derive a first-order correction for the eigenvalues using only the neutrally buoyant eigenvalue problem. This eliminates the need to re-solve the eigenvalue problem at each stratification level. For ideal-gas boundary layers, the framework accurately predicts how stable and unstable stratification modifies Tollmien-Schlichting waves, from growth rates and eigenfunctions to $N$-factors, holding across a wide range of Prandtl numbers, temperature ratios, and Mach numbers. Notably, the buoyancy sensitivity varies strongly with Prandtl number, revealing that for a given Richardson number, buoyancy can switch from destabilising to stabilising depending on the fluid. Beyond ideal-gas conditions, we apply the first-order buoyancy correction to strongly stratified boundary layers with supercritical fluids, where the phase relationship between density and velocity perturbations determines whether buoyancy stabilises or destabilises the underlying instability. The resulting $N$-factors demonstrate, for the first time, that buoyancy significantly affects transition predictions under pseudo-boiling conditions.

Figures (14)

• Figure 1: (a) Base-flow profiles of streamwise velocity $\bar{u}/u^*_\infty$, temperature $\bar{T}^*/T^*_\infty$, and density $\bar{\rho}^*/\rho^*_\infty$ for $T^*_\mathit{w}/T^*_\infty=1.01$ over the dimensionless wall-normal coordinate $y^*/\delta^*$. The black arrow indicates the direction of gravity. The velocity and thermal boundary-layer thicknesses, $\delta_{99}$ and $\delta_t$, are indicated, respectively. (b) Neutral-stability curves in the $Re$--$F$ plane for $T^*_\mathit{w}/T^*_\infty=1.01$ at $Ri=[-0.04,0,0.04]$. The black dotted line shows the neutral stability of the Blasius profile.
• Figure 2: Contours of (a) $\Im\{C_0\}$ and (b) $\Re\{C_0\}$ in the $Re$--$F$ plane. The black solid line indicates the neutral-stability curve where $\Im\{\alpha_0\}=0$.
• Figure 3: Neutral-stability curves in the $Re$--$F$ plane for stably and unstably stratified cases at $Ri=[-0.1,-0.04,0.04]$. Symbols ($\circ$) show results from the buoyant eigenvalue problem \ref{['eq:residual_wbuo']}; solid lines denote the first-order correction \ref{['eq.dadRi2']} evaluated with $C_0$; dotted lines depict the first-order correction \ref{['eq.dadRi2']} evaluated with $\bar{C}_0$. The neutrally buoyant case at $Ri=0$ is indicated with a black dashed line. The black pentagram indicates the location at which the eigenfunctions are extracted in Appendix \ref{['sec:appB']}.
• Figure 4: (a) Growth rate and (b) phase speed as functions of $Re$ at $F=45 \times 10^{-6}$ for stably and unstably stratified cases at $Ri=[-0.1,-0.04,0.04]$. Symbols ($\circ$) show results from the buoyant eigenvalue problem \ref{['eq:residual_wbuo']}; solid lines denote the first-order correction \ref{['eq.dadRi2']} evaluated with $C_0$; dotted lines depict the first-order correction \ref{['eq.dadRi2']} evaluated with $\bar{C}_0$. The neutrally buoyant case at $Ri=0$ is indicated with a black dashed line.
• Figure 5: $N$-factor envelopes at $Ri=[-0.1,-0.04,0,0.04]$. Symbols ($\circ$) show results from the buoyant eigenvalue problem \ref{['eq:residual_wbuo']}; solid lines denote the first-order correction \ref{['eq.dadRi2']} evaluated with $C_0$; dotted lines depict the first-order correction \ref{['eq.dadRi2']} evaluated with $\bar{C}_0$. The $N$-factor of the neutrally buoyant case is indicated with a black dashed line. The location of maximum amplification rate, $\max\{-\Im\{\alpha_0\}/Re\}$, is indicated with a black pentagram.